民眾對於產品品質要求逐漸提高，故製造商除了要設計出高可靠度的產品外，也要找出生產後容易失效的產品，因此製造商會藉由預燒試驗來找出早期失效的產品，預燒試驗是將待測品放入預燒爐中進行測試，藉由高溫、高電壓、高電流等環境消除產品浴缸曲線中早夭期中的不良品，若產品在時間內通過預燒試驗，才得以販售給消費者，因此本研究希望能夠決定出最佳的預燒時間。
而經過預燒後的產品，販售到市面上後，消費者難以判斷品質的好壞，因此保固常扮演一個重要的角色，保固不僅能保障消費者與廠商的權益，且也是一個相當好的行銷手段。但在保固期間最常遇到產品發生失效的情形，此時製造商需要負責進行維修產品的義務，因此產品售出後的保固費用成為生產成本中必須考慮的一部分，但如果保固期太短，消費者對於該產品品質產生不信任；而保固期太長，製造商又需要負擔較多的維修費用。因此如何決定適當的保固長度也是本研究希望探討的問題。
本研究採用虛擬年齡模型去表達進行不完美維修後的使用狀態，不完美維修的程度會產生不同的年齡縮減因子，將不同的年齡縮減因子帶入失效率中，透過蒙地卡羅模擬去模擬在不同年齡縮減因子失效率下發生的失效時間點，以此計算在整個產品生命週期中經歷預燒、保固及過保固後的平均失效次數。
而在過去預燒及保固的文獻中並沒有探討到考慮到不完美維修的情形，因此本研究將考慮產品在符合浴缸曲線下計算預燒成本，保固期間不完美維修的成本及過保固後一旦失效產生的懲罰成本三部分，決定出最佳預燒時間、最佳保固時間及最佳不完美維修效果，以最小化整體總成本。

In this study, an optimization model is developed to investigate the optimal burn-in time, warranty length and age reduction factor of a product from a manufacturer’s perspective, so that the mean of total product servicing cost is minimized. It is assumed that when the product fails, we conduct minimal repair during burn-in time. The products after burn-in are sold to the market, it is difficult for consumers to judge quality. Therefore, the warranty often plays an important role. If the product during warranty fails before bathtub curve first change point, then we conduct minimal repair. If the product fails after bathtub curve first change point, then we conduct imperfect repair. In this study, a virtual age model is used to express the status after imperfect repair. The degree of imperfect repair will generate different age reduction factors, and bring different age reduction factors into the failure rate. Out-of-warranty, if the product fails before its useful life limits, it causes customer dissatisfaction and incurs a penalty cost for the manufacturer. And Monte Carlo simulations were used to simulate the time-to-failures at different age-reducing factor failure rates to calculate the average number of failures after burn-in, warranty, and post-warranty throughout the product life cycle. We provide a numerical example to determine the optimal burn-in time, optimal warranty length, and optimal imperfect repair degree to minimize total cost. A sensitivity analysis is conducted to evaluate the effect of model parameters on the optimal solution.

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